Optimal control of two-dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone

Abstract

Summary Our work is devoted to an optimal control problem for two‐dimensional parabolic partial differential equations(PDEs) and its application in engineering sciences. An adjoint problem approach is used for analysis of the Fréchet gradient of the cost functional, and we prove the gradient is Lipschitz continuous. An improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm we proposed is studied. Results of some computational experiments obtained by the conjugate gradient algorithm are illustrated. The results show that the improved conjugate gradient algorithm is effective. Copyright © 2016 John Wiley & Sons, Ltd.

First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

“…With the rapid development of technologies, more and more researchers are using the FOTD approach and different algorithms to solve the PDE-constrained optimization problems for various research fields. For instance, Wang et al [8] use the FOTD approach to solve a two-dimensional parabolic PDE-constrained optimal control problem, which is applied for the cooling process of steel billets in continuous casting secondary cooling zones. Luo et al [9] use the FOTD strategy to solve the 2-dimensional PDE optimal control problem to obtain the reference values of the optimal furnace zone temperatures for the reheating furnace.…”

GPU-Based Model Predictive Control of Nonlinear Parabolic Partial Differential Equations System and Its Application in Continuous Casting

“…Several optimization methods have been extensively developed to solve this problem, such as gradient method [20], conjugate gradient method [21], [22], Levenberg-Marquardt method (LM) [18], [23], Cao method [24], and Lanbweber method [25]. The basic concept of these optimization methods is to update h for each iteration via the gradient of cost function or the value of cost function.…”

Dual strategy for 2‐dimensional PDE optimal control problem in the reheating furnace

“…It is easy to prove that the PM is globally convergent under the modified Armijo‐type line search. In another work, our colleague Yuan Wang discussed the sufficient descent property and gave a detailed proof of the global convergence. Hence, we ignore the proof process here.…”

“…The method was based upon the iterative solution of an inverse problem with the use of the adjoint equations to make the method computationally feasible. Wang et al used the adjoint method for an optimal control problem of 1D and 2D parabolic PDEs, which describes a cooling process in continuous casting. Based on the adjoint method, Stoll and Wathen presented an all‐at‐once approach where they solved for all time‐steps of the discretized unsteady Stokes problem at once.…”

“…In general, this approach is used to solve the 1D inverse parabolic problem. () To the authors' knowledge, there have been several studies that use the idea of adjoint equations in the areas of seismology, biomechanics,() continuous casting process,() fluid mechanics, and some others. For instance, Plessix introduced the adjoint‐state method to geophysical applications by defining some adjoint‐state variables that are solutions of a linear system.…”